establish retention probabilities

Establish retention

probabilities

Adjust resident and extension student credit

hour ratios by residency and CIP level

Adjust assigned housing, meal plans,

and parking ratiosOperational areas establish tentative goals

for new students

Review revenues, student credit hours, and housing in light

of tentatively established goals

Monitor progress toward goals, adjust model and

report progress

Projections Flow ModelAn Iterative Process

1

Operational areas establish goals

Establish new student goal forecasts/

ranges

Undertake new student goal setting and what if analyses

with operational areas

2

3

4

5

67

8

9

Establish retention probabilities1

Establish retention

probabilities





for new students




report progress


1


Establish new student goal

forecasts/ ranges



23

4

5

67

8

9

Establish new student goal forecasts/ranges2

Establish retention

probabilities





for new students




report progress


1



forecasts/ ranges



23

4

5

67

8

9

Undertake new student goal setting and what if analyses with operational areas3



Establish retention

probabilities

Adjust resident and extension student credit hour ratios by residency

and CIP level



for new students




report progress


1



forecasts/ ranges

23

4

5

67

8

9

Adjust resident and extension student credit hour ratios by residency and CIP level4



Establish retention

probabilities





for new students




report progress


1



forecasts/ ranges

23

4

5

67

8

9

Adjust assigned housing, meal plans, and parking ratios5



Establish retention

probabilities





for new students




report progress


1



forecasts/ ranges

23

4

5

67

8

9

Operational areas establish tentative goals for new students6



Establish retention

probabilities





for new students

Review revenues, student credit

hours, and housing in light of tentatively

established goals


report progress


1



forecasts/ ranges

23

4

5

67

8

9

Review revenues, student credit hours, and housing in light of tentatively established goals7



Establish retention

probabilities





for new students




report progress


1



forecasts/ ranges

23

4

5

67

8

9

Operational areas establish goals8

Establish retention

probabilities





for new students




report progress


1



forecasts/ ranges

23

4

5

67

8

9Undertake new

student goal setting and what if analyses


Monitor progress toward goals, adjust model and report progress9

Establish retention

probabilities





for new students




report progress


1


Establish new student goal forecasts/

ranges



23

4

5

67

8

9

Time Series and Forecasting

• Time-series data describes the movement of a variable over time.

• Forecasting – Making quantitative estimates about the likelihood of future events which is developed on the basis of past and current information.

Stochastic Time Series Models• Regular least squares regression models

are deterministic and do not rely on the randomness of the errors.

• Stochastic models are random process models, where we are assuming that the dependent variables are drawn randomly from a probability distribution.

• These models capture the characteristics of the series’ randomness.

Time Series Components• Trend – The upward and downward movement that

characterizes a time series of a period of time. • Seasonal Variations – Periodic patterns in a time series

that complete themselves within a calendar year and are then repeated on a yearly basis.

• Cycle – Recurring up and down movements around trend levels (ex. business cycles).

• Irregular Fluctuations – Erratic movements in a time series that follow no recognizable or regular pattern (what is ‘left over’ after trend, cycle, and seasonal variations have been accounted for).

Typical Forecast Errors

Serial Correlation• Common in time series data.• Occurs when the errors corresponding to different

observations are correlated.• Violates the assumption that the errors are independent

over time.• Ordinary least-squares estimators will have lower

efficiency.• Detection:

– Durbin-Watson Test – Tests the null hypothesis that no serial correlation is present.

• Correction:– Generalized differencing.– Adding an autoregressive (ρ) process

Stationarity• Stationary Time Series – A time

series where the underlying stochastic process that generated the series can be assumed to be invariant with respect to time.

• If the characteristics of the stochastic process change over time then the series is nonstationary and the coefficients will not be fixed. Nonstationary processes can often be converted into stationary processes (typically through differencing).

• The residuals for a stationary time series should resemble white noise, with a mean of zero.

Sample Autocorrelation Function (SAC)• Measures the linear relationship between time series observations

separated by a lag of k time units.– Values close to +1 indicate that observations separated by a lag of k time units

have a strong tendency to move together in a linear fashion with a positive slope, and values close to -1 move together with a negative slope.

• Behavior of the SAC– Cut-Off – A spike at lag k exists in the SAC if the value is statistically large.

• Regular peaks in the SAC are indicative of seasonality.– Dies Down – SAC does not cut-off, but rather decreases in a “steady fashion.”

• A damped exponential fashion (with or without oscillation)• A damped sine-wave fashion• A fashion dominated by either one of or a combination of the two above

– If the SAC either cuts off fairly quickly or dies down fairly quickly then the time series should be considered stationary.

• If there are spikes then the series may not be stationary at the seasonal level– If the SAC dies down extremely slowly then the time series should be considered

nonstationary.

ARIMA(p,d,q) Models• Objective is to explain the movement of a time series by relating it to its own

past values and to a weighted sum of current and lagged random disturbances.

• Assumes stationarity and normally distributed error terms.• Has a moving average component and autoregressive component.• Moving Average Models – Used to smooth out short-term fluctuations in

order to highlight long-term variation due to trends, cycles, and seasonality.– Generated by a weighted average of random disturbances going back q periods.– Has a limited memory of past periods, which suggests that moving average

models by themselves are poor forecasting models.• Autoregressive Models - Tells us how much correlation there is (and how

much interdependency there is) between neighboring data points in the series.– Generated by a weighted average of past observations going back p periods,

together with a random disturbance in the current period.– If stationary then its mean must be finite and invariant with respect to time.– Has an infinite memory, which means that the current value of the process

depends on all past values, although the magnitude of this dependence declines with time.

ARIMA(p,d,q) Model Identification

• Consists of a four-step iterative process:– Tentative Identification – Historical data are used to

tentatively identify an appropriate model.– Estimation – Historical data are used to estimate the

parameters of the tentatively identified model.– Diagnostic Checking – Various diagnostics are used

to check the adequacy of the tentatively identified model and, if need be, to suggest an improved model, which is then regarded as a new tentatively identified model.

– Forecasting – Once a final model is obtained, it is used to forecast future time series values.

ARIMA(p,d,q) Model Identification (Continued)

Model SAC SPAC

Moving average of order q Cuts off after lag q Dies down

Autoregressive of order p Dies down Cuts off after lag p

Mixed autoregressive-moving average of order (p,q) Dies down Dies down

• The first problem is to determine the degree of homogeneity d, that is, the number of times the series must be differenced to produce a stationary series. To do this we difference until the SAC shows stationarity.

• To determine the orders of the autoregressive and moving average components we use both the SAC and SPAC (sample partial autocorrelation function). The SPAC operates in a similar manner to the SAC.– Spikes in the SAC are indicative of moving average terms.– The SPAC can be used for guidance in determining the order of the

autoregressive portion of the process.

Seasonal ARIMA Modeling• We define lags L, 2L, 3L, etc. as exact seasonal lags.• We define lags L-2,L-1,L+1,L+2,2L-2,2L-1,2L+1,2L+2, etc as near

seasonal lags.• Seasonal Moving Average Model of Order Q:

– SAC has nonzero autocorrelations at lags L,2L,3L,…,QL and zero autocorrelations elsewhere.

– SPAC dies does at seasonal lags L,2L,3L,…• Seasonal Autoregressive Model of Order P:

– SAC dies down at seasonal lags L,2L,3L,…– SPAC has nonzero partial autocorrelations at lags L,2L,…,PL and zero

partial autocorrelations elsewhere.• To correctly identify a model we would use the SAC and SPAC at

the nonseasonal level to tentatively identify a nonseasonal model, use the SAC and SPAC at the seasonal level to tentatively identify a seasonal model, and then combine the nonseasonal and seasonal models to arrive at an overall tentatively identified model.

Diagnostic Checking• The first step in diagnostic checking of the model would be to

calculate the SAC for the residuals of the estimated ARIMA(p,d,q) model and determine if those residuals appear to be white noise. If they do not, then a new model specification can be tried.– The Box and Pierce Q statistic and Ljung-Box statistic can both be used

to test model adequacy.• Can also pick the best model based on the basis of forecasting

performance.– We can do this by looking for the model with the lowest mean-square-

error forecasts.• If the model has an autoregressive component stationarity can be

checked by looking at the autoregressive parameters:– First-order: -1<Ф1<1– Second-order: Ф1 + Ф2<1 and Ф2-Ф1<1 and -1<Ф2<1

Other Considerations• Time series data is usually nonstationary and

therefore should be transformed.• Even when the data appears stationary and the

trend is not statistically significant it still might be worthwhile to difference the data in order to produce more accurate forecasts.

• Forecasting is an art as much as it is a science. Even though the SAC and SPAC may lead to a clearly defined model there may be other models that are more practical.

• The more data the better.

SAS vs. SPSS

SAS• More fine grained

control over modeling

• More control over data manipulation

• Better integration with other software

SPSS• Easier to use• Easier to read

output

Using Delaware Cost of Instruction (UNC Funding Model Example)

Student Credit Hours (SCH) per Instructional Position

Funding Category (CIP Codes) Undergraduate Masters Doctoral

Category 1 708.64 169.52 115.56

Category 2 535.74 303.93 110.16

Category 3 406.24 186.23 109.86

Category 4 232.25 90.17 80.91

12-Cell Matrix of Instructional Level and Disciplinary Instructional Areas

CIP Program Title Funding Category

09 Communications 1

23 English Language and Literature/Letters 1

27 Mathematics 1

38 Philosophy and Religion 1

42 Psychology 1

43 Protective Services 1

45 Social Sciences and History 1

54 History 1

13 Education 2

16 Foreign Languages and Literatures 2

19 Home Economics 2

30 Mulit/Interdisciplinary Studies 2

31 Parks, Recreation, Leisure and Fitness Studies 2

52 Business Management and Administrative Services 2

03 Conservation and Renewable Natural Resources 3

11 Computer and Information Sciences 3

15 Engineering-Related Technologies 3

26 Biological Sciences/Life Sciences 3

40 Physical Sciences 3

44 Public Administration and Services 3

50 Visual and Performing Arts 3

51 Health Professions and Related Sciences 3

14 Engineering 4

51 Nursing 4

Student Credit Hour Projections

• Applied averages of SCH to headcount by credit type and residence to projected headcounts to project future SCHs.

• Applied averages of SCH by funding category to overall SCHs to projected SCHs to determine projections by funding category.

establish retention probabilities

Documents

student credit hours

new studentsreview revenues

tentative goals

cip leveladjust

meal plans